Other approaches use differential inequalities, comparison
principles, interval arithmetic, and combinations thereof, and compute
interval over-approximations
[1]}, [2]}, [3]}, [4]}, [5]}.
While these techniques may allow for uncertain coefficients \(A\)  and
\(B\)  in REF  [4]} or even for
nonlinear dynamics
[1]}, [2]}, [3]}, [5]}, they are all conservative, i.e.,
arbitrarily precise over-approximations of reachable tubes cannot be
obtained, and the methods in
[1]}, [2]}, [3]}, [4]}
additionally suffer from both shortcomings mentioned in our discussion
of ellipsoidal techniques.
Finally, the reachable tube can also be characterized as a sublevel
set of the viscosity solution of a partial differential equation
called Hamilton-Jacobi-Bellman equation
[15]}. However, solving the latter numerically
is avoided in practice as this would require discretizing the state
space and so the computational effort would scale exponentially with
the state space dimension.
